Final answer:
To calculate the refractive index of the cladding, one must manipulate the equation for the critical angle and acceptance angle using Snell's Law. By rearranging to solve for the cladding's refractive index and substituting the given values, it is found that the refractive index of the cladding is approximately 1.308. Option B is correct.
Step-by-step explanation:
The student's question asks for the calculation of the refractive index of the cladding of a fiber cable, given an acceptance angle of 30° and a core index of refraction of 1.4.
To solve this problem, we use Snell's Law and the concept of critical angle. The critical angle (\( \theta_c \)) can be determined by using the equation \( \sin(\theta_c) = \frac{n_{cladding}}{n_{core}} \) where \( n_{core} \) is the index of refraction of the fiber core and \( n_{cladding} \) is the index of refraction of the cladding.
We also know the relationship between the acceptance angle (\( \theta_a \)) and the critical angle: \( \theta_a = \sin^{-1}(\sqrt{n_{core}^2 - n_{cladding}^2}) \).
Working from this, we can rearrange the equation to solve for \( n_{cladding} \), giving us:
\( n_{cladding} = \sqrt{n_{core}^2 - (\sin(\theta_a))^2} \)
Plugging the given values into our equation:
\( n_{cladding} = \sqrt{1.4^2 - (\sin(30\degree))^2} \)
\( n_{cladding} = \sqrt{1.96 - 0.25} \)
\( n_{cladding} = \sqrt{1.71} \)
After calculating the square root, we find that the refractive index of the cladding is:
\( n_{cladding} \approx 1.308 \)
Therefore, the correct answer is B. 1.308.